Question

Could a system of two linear equations ever have an infinite number of solutions?If so, how could you tell by looking at their equations? How could you tell from the graph?If not, explain why.

Answer 1

Yes. Two linear equations in x and y can have infinitely many solutions when they represent the exact same line.

How to tell from the equations
- Write each equation in standard form ax + by = c. The system has infinitely many solutions exactly when the coefficients are proportional:
a1/a2 = b1/b2 = c1/c2
(i.e., one equation is a constant multiple of the other).
- In slope–intercept form y = mx + b, this is the case when the slopes and y-intercepts are equal: m1 = m2 and b1 = b2.
- (More generally with matrices: infinite solutions occur when the coefficient matrix is singular and the augmented matrix has the same rank as the coefficient matrix.)

How to tell from the graph
- The two lines coincide (are the same line). Graphically they lie on top of each other, so every point on the line satisfies both equations, giving infinitely many intersection points.

Quick contrast
- If a1/a2 = b1/b2 but c1/c2 is different, the lines are parallel and distinct → no solution.
- If a1/a2 ≠ b1/b2, the lines intersect in exactly one point → one solution.

Example
- 2x + 4y = 6 and x + 2y = 3 are the same line (multiply the second by 2), so the system has infinitely many solutions.